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The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak1,2, K.A. Penson1, and A.I. Solomon1
1Université Pierre et Marie Curie, Laboratoire de Physique Théorique des Liquides CNRS URA 7600, Tour 16, 5ièmeétage, 4, place Jussieu, 75252 Paris Cedex 05, France
{blasiak, penson}@lptl.jussieu.fr, a.i.solomon@open.ac.uk
2H. Niewodniczański Institute of Nuclear Physics, ul.Eliasza Radzikowskiego 152 31-342 Kraków, Poland
Annals of Combinatorics 7 (2) p.127-139 June, 2003
AMS Subject Classification: 81R05, 81R15, 81R30, 47N50
For any function F(x) having a Taylor expansion we solve the boson normal ordering problem for , with r, s positive integers, =1, i.e., we provide exact and explicit expressions for its normal form , where in all a's are to the right. The solution involves integer sequences of numbers which, for r, s1, are generalizations of the conventional Bell and Stirling numbers whose values they assume for r=s=1. A complete theory of such generalized combinatorial numbers is given including closed-form expressions (extended Dobinski-type formulas), recursion relations and generating functions. These last are special expectation values in boson coherent states.
Keywords: boson normal order, Bell numbers, Stirling numbers, coherent states


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